How To Calculate Averages
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The arithmetic mean
“The average” is doing a lot of work in most sentences, and people usually mean the arithmetic mean—add everything up, divide by the count. But the median and the mode are also averages, and in skewed data they often describe the typical case better than the mean does. The average household income in the United States is higher than the median household income because a handful of billionaires pull the mean upward; report the mean and you’ll mislead people about the “middle” household. Weighted averages appear everywhere grades are calculated, geometric means show up in finance and biology, and harmonic means matter when averaging rates. This guide covers the three main averages and when to use each, weighted averages, how outliers distort the mean, and when geometric or harmonic means are the right choice.
The median
Sum the values, divide by the count. For test scores 80, 90, 70, 100: sum = 340, count = 4, mean = 85. This is what most people call “the average.” It’s the balance point of the data—if you placed weights at each value on a number line, the mean is where the line balances.
The mode
Sort the values and pick the middle one. For an even count, take the mean of the two middle values. The median is insensitive to outliers—changing the largest value from 100 to 1,000,000 doesn’t move the median at all. This makes it the right choice for skewed distributions like income, house prices, and web page load times.
Mean vs. median vs. mode: when to use each
The value that appears most often. There can be zero modes (all values unique), one mode (unimodal), or multiple modes (bimodal, multimodal). Mode is useful for categorical data (“most common shirt size”) where mean and median don’t make sense, and for spotting clustering in numeric data.
Weighted averages
When some values count more than others, multiply each by its weight, sum, then divide by the total weight. Course grades are classic: homework 20%, midterm 30%, final 50%. A student with 85/100/70 on those gets (85×0.2 + 100×0.3 + 70×0.5) / 1 = (17 + 30 + 35) = 82.
The effect of outliers
A single extreme value can wildly distort the mean. Consider seven salaries: six at $50,000 and one CEO at $2,000,000. The mean salary is $328,571. The median is $50,000. The mean is mathematically correct but communicatively misleading. When reporting averages to humans, check the shape of the distribution first—if there’s a long tail, the median is more honest.
Geometric mean
An average alone is usually insufficient. Two datasets can have identical means with wildly different spreads: {`{49, 50, 51}`} and {`{0, 50, 100}`} both average 50, but you’d prefer to know you’re walking into the first one. Report an average alongside a measure of spread—standard deviation, interquartile range, or min/max.